On Graphs With Small Ramsey Numbers,II

There exists a constant C such that for every d-degenerate graphs G ₁ and G ₂ on n vertices, Ramsey number R(G ₁,G ₂) is at most Cn, where is the minimum of the maximum degrees of G ₁ and G ₂.* The work of this author was supported by the grants 99-01-00581 and 00-01-00916 of the Russian Foundation for Fundamental Research and the Dutch-Russian Grant NWO-047-008-006. The work of this author was supported by the NSF grant DMS-9704114.     

Regressions and monotone chains II: The poset of integer intervals

A regressive function (also called a regression or contractive mapping) on a partial order P is a function mapping P to itself such that (x)x. A monotone k-chain for is a k-chain on which is order-preserving; i.e., a chain x ₁<...ksuch that (x ₁)...(x_k). Let P _nbe the poset of integer intervals {i, i+1, ..., m} contained in {1, 2, ..., n}, ordered by inclusion. Let f(k) be the least value of n such that every regression on P _nhas a monotone k+1-chain, let t(x,j) be defined by t(x, 0)=1 and t(x,j)=x ^t(x,j–1). Then f(k) exists for all k (originally proved by D. White), and t(2,k) < f(K) <t( + _k, k) , where _k 0 as k. Alternatively, the largest k such that every regression on P _nis guaranteed to have a monotone k-chain lies between lg^*(n) and lg^*(n)–2, inclusive, where lg^*(n) is the number of appliations of logarithm base 2 required to reduce n to a negative number. Analogous results hold for choice functions, which are regressions in which every element is mapped to a minimal element.     

The local nature of Δ-coloring and its algorithmic applications

Given a connected graphG=(V, E) with |V|=n and maximum degree such thatG is neither a complete graph nor an odd cycle, Brooks' theorem states thatG can be colored with colors. We generalize this as follows: letG-v be -colored; then,v can be colored by considering the vertices in anO(log n) radius aroundv and by recoloring anO(log n) length augmenting path inside it. Using this, we show that -coloringG is reducible inO(log³ n/log) time to (+1)-vertex coloringG in a distributed model of computation. This leads to fast distributed algorithms and a linear-processorNC algorithm for -coloring.A preliminary version of this paper appeared as part of the paper Improved Distributed Algorithms for Coloring and Network Decomposition Problems, in theProceedings of the ACM Symposium on Theory of Computing pages 581–592, 1992. This research was done when the authors were at the Computer Science Department of Cornell University. The research was supported in part by NSF PYI award CCR-89-96272 with matching funds from UPS and Sun Microsystems.     

On the generalized twisted field planes of characteristic 2

Let be a non-Desarguesian semifield plane of order 2 ⁿ 2⁶, and letG be the autotopism group relative to an autotopism triangle . We prove that ifG acts transitively on the non-vertex points on a side of , then is a generalized twisted field plane. A characterization of the generalized twisted field planes of characteristic 2 is also given.Research supported in part by NSF Grants RII-9014056, component IV of the EPSCoR of Puerto Rico grant and ARO grant for Cornell MSI.Research supported in part by NSF Grant No. DMS-9107372.     

Optimal growth rates for a viscous Hamilton-Jacobi equation

The growth of the L^m-norm, m [1,], of non-negative solutions to the Cauchy problem _t u – u = |u| is studied for non-negative initial data decaying at infinity. More precisely, the function is shown to be bounded from above and from below by positive real numbers. This result indicates an asymptotic behaviour dominated by the hyperbolic Hamilton-Jacobi term of the equation. A one-sided estimate for ln u is also established.     

Limit distributions of U-statistics resampled by symmetric stable laws

Summary If (Y _i) and (V _i) are independent random sequences such thatY _i are i.i.d. random variables belonging to the normal domain of attraction of a symmetric -stable law, 0<<2, andV _i are i.i.d. random variables, then the limit distributions of U-statistics , coincide with the probability laws of multiple stochastic integralsX ^d f = ... f (t ₁, ... ,t _d)dX(t _d) with respect to a symmetric -stable processX(t).The research was originated during author's visit at ORIE, Cornell University     

The Phase Space of One Generalized Model by Oskolkov

We demonstrate that the phase space of the Cauchy–Dirichlet problem for the equation u _t -u _t =u-K(u)+f is a simple Banach C -manifold.     

Combinatorial decompositions and homogeneous geometrical processes

This paper considers line processes and random mosaics. The processes are assumed invariant with respect to the group of translations ofR ². An expression for the probabilities ,k=0, 1, 2,... to havek hits on an interval of lengtht taken on a typical line of direction (the hits are produced by other lines of the process) is obtained. Also, the distribution of a length of a typical edge having direction in terms of the process {P _i , _i } is found, hereP _i is the point process of intersections of edges of the mosaic with a fixed line of direction and the mark _i is the intersection angle atP _i . The method is based on the results of combinatorial integral geometry.     

Characterizations of Buchsbaum complexes

Letk be a field and an abstract simplicial complex with vertex set . In this article we study the structure of the Ext modules Ext _a ⁱ (A/m _(l ,k[]) of the Stanley-Reisner ringk[] whereA=k[x ₁,...,x _n ] andm _l =(x _l ¹ ,...,x _l ⁿ ). Using this structure theorem we give a characterization of Buchsbaumness ofk[] by means of the length of the modules Ext _A ⁱ (A/m _l ,k[]). That isk[] is Buchsbaum if and only if for allik[], the length of the modules Ext _A ⁱ (A/m _l ,k[]) is independent ofl.     

The normalized matching property from the generalized Macaulay theorem

IfA is a family ofk-element subsets of a finite setM having elements of several different types (i.e., amultiset) and A is the set of all (k–1)-element subsets ofM obtainable by removing a single element ofM from a single member ofA, then, according to the well known normalized matching condition, the density ofA among thek-element subsets ofM never exceeds the density of A among the (k–1)-element subsets ofM. We show that this useful fact can be regarded as yet another corollary of the generalized Macaulay theorem.     

On the homogeneity of additive mappings

LetK be a ring with an identity 1 0 andM, L two unitaryK-modules. Then, for any additive mappingf:M L, the setH _f :={ K f(x)=f(x) for allx M} forms a subring ofK, the homogeneity ring off. It is shown that, forM {0},L {0} and any subringS ofK for whichM is a freeS-module, there exists an additive mappingf:ML such thatH _f =S. This result is applied to the four Cauchy functional equations, and it leads also to an answer to the question as to whether it is possible to introduce onM a multiplication ·:M × M M makingM into a ring but not into aK-algebra.     

Lattice Polytopes Associated to Certain Demazure Modules of sl n + 1

Let w be an element of the Weyl group of sl _{n + 1}. We prove that for a certain class of elements w (which includes the longest element w₀ of the Weyl group), there exist a lattice polytope R ^l(w) , for each fundamental weight _i of sl _{n + 1}, such that for any dominant weight = _{i = 1} ⁿ a _{i i} , the number of lattice points in the Minkowski sum _w = _{i = 1} ⁿ a _{i i} ^w is equal to the dimension of the Demazure module E _w (). We also define a linear map A ^w : R ^l(w) P _Z R where P denotes the weight lattice, such that char E _w () = e e^–A(x) where the sum runs through the lattice points x of ^w .     

Quadratic spline interpolation on uniform meshes

The interpolation problem at uniform mesh points of a quadratic splines(x _i)=f _i,i=0, 1,...,N ands(x ₀)=f₀ is considered. It is known that s–f=O(h ³) and s–f=O(h ²), whereh is the step size, and that these orders cannot be improved. Contrary to recently published results we prove that superconvergence cannot occur for any particular point independent off other than mesh points wheres=f by assumption. Best error bounds for some compound formulae approximatingf _i andf _i ⁽³⁾ are also derived.     

Convergence of Riemann sums for stochastic integrals

Summary Let b be a Brownian motion and f a function in L ²[0,1]. If is a partition of [0,1], denote by f the step function obtained by replacing f by its mean values in each subinterval. As becomes fine, the martingale f db converges to fdb in L ² but not necessarily almost surely. We determine precisely which Lipschitz conditions on f imply a.s. convergence. A similar thing is done for non-anticipating random functions.     

On construction ofk-wise independent random variables

A 0–1probability space is a probability space (, 2,P), where the sample space -{0, 1} ⁿ for somen. A probability space isk-wise independent if, whenY _i is defined to be theith coordinate or the randomn-vector, then any subset ofk of theY _i 's is (mutually) independent, and it is said to be a probability spacefor p ₁,p ₂, ...,p _n ifP[Y _i =1]=p _i .We study constructions ofk-wise independent 0–1 probability spaces in which thep _i 's are arbitrary. It was known that for anyp ₁,p ₂, ...,p _n , ak-wise independent probability space of size always exists. We prove that for somep ₁,p ₂, ...,p _n [0,1],m(n,k) is a lower bound on the size of anyk-wise independent 0–1 probability space. For each fixedk, we prove that everyk-wise independent 0–1 probability space when eachp _i =k/n has size (n _k ). For a very large degree of independence —k=[n], for >1/2- and allp _i =1/2, we prove a lower bound on the size of . We also give explicit constructions ofk-wise independent 0–1 probability spaces.This author was supported in part by NSF grant CCR 9107349.This research was supported in part by the Israel Science Foundation administered by the lsrael Academy of Science and Humanities and by a grant of the Israeli Ministry of Science and Technology.     

A Note on Common Zeroes of Laplace–Beltrami Eigenfunctions

Let u+u=v+v= 0, where isthe Laplace–Beltrami operator on a compact connected smoothmanifold M and > 0. If H ¹(M) = 0then there exists pM such that u(p)=v(p) = 0 For homogeneous M,H ¹(M) 0 implies the existence of a pair u,v as above that has no common zero.     

Symplectic Toric Space Associated to Triangle Inequalities

Let M _n be the moduli space of spatial polygons with n edges. An open dense subset of M _n admits a T ^n–3 -action, although this action does not extend to M _n . The action together with a symplectic structure on M _n naturally defines a convex polytope _n in ^n–3. In this paper, from M _n , we construct a singular symplectic toric manifold with _n as the image of the moment map.     

Infinite highly arc transitive digraphs and universal covering digraphs

A digraph (that is a directed graph) is said to be highly arc transitive if its automorphism group is transitive on the set ofs-arcs for eachs0. Several new constructions are given of infinite highly arc transitive digraphs. In particular, for a connected, 1-arc transitive, bipartite digraph, a highly arc transitive digraphDL() is constructed and is shown to be a covering digraph for every digraph in a certain classD() of connected digraphs. Moreover, if is locally finite, thenDL() is a universal covering digraph forD(). Further constructions of infinite highly arc transitive digraphs are given.The second author wishes to acknowledge the hospitality of the Mathematical Institute of the University of Oxford, and the University of Auckland, during the period when the research for this paper was doneResearch supported by the Australian Research Council